Time Series Checking with Fuzzy Interval Temporal Logics

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Temporal Logics

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Willem Conradie1[0000−0001−9906−4132], Dario Della Monica2[0000−0001−9743−665X], Emilio Mu˜noz-Velasco3[0000−0002−0117−4219],

Guido Sciavicco4[0000−0002−9221−879X], and Ionel Eduard Stan4,5[0000−0001−9260−102X]

1 School of Math., Wits University, South Africa willem.conradie@wits.ac.za

2 Dept. of Math., Comp. Sci., and Phys., University of Udine, Italy dario.dellamonica@uniud.it

3 Dept. of App. Math., University of M´alaga, Spain ejmunoz@uma.es

4 Dept. of Math. and Comp. Sci., University of Ferrara, Italy {guido.sciavicco|ioneleduard.stan}@unife.it

5 Dept. of Math., Phys., and Comp. Sci., University of Parma, Italy

Abstract. Model checking is a very well-known problem, with many practical applications. A possible variation of such a problem in the interval logic setting is the so-called finite model checking, that consists of verifying an interval temporal logic formula, typically of Halpern and Shoham’s logic of Allen’s relations HS, on a fully represented finite interval model. Multivariate time series are collections of temporally ordered sets of values, and they allow to describe a variety of situations, such as the medical history of an hospitalized patient or the sensor values during a plane flight. In this paper we argue how the recently introduced fuzzy generalization of interval temporal logic is a suitable language in which interesting properties of a multivariate time series can be expressed and checked, and we define, solve, and discuss the complexity of the multivariate time series fuzzy interval logic checking problem.

Keywords: Model checking·Time series·Fuzzy logic.

1 Introduction

A temporal variable is a variable whose value changes over time. A time series is a set of temporal variables. They can be univariate, if only one temporal

?Copyright c2020 for this paper by its authors. Use permitted under Creative Com- mons License Attribution 4.0 International (CC BY 4.0). The authors acknowl- edge the partial support by the Italian INDAM-GNCS projectStrategic Reasoning and Automated Synthesis of Multi-Agent Systems, by the National Spanish project PGC2018-095869-B-I00Fuzzy, Logic-based, and Algebraic tools for Information Re- sources, and by the Andalucia Region project UMA2018-FEDERJA-001Mathemat- ics for the Imperfect Information.

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variable is involved, or multivariate, otherwise. Each variable of a multivariate time series is an ordered collection ofnnumeric values, instead of a single value.

Multivariate time (or temporal) series emerge in many application contexts: the temporal history of some hospitalized patients can be described by the time series of the values of their temperature, blood pressure, and oxygenation; the pronunciation of a word in sign language can be described by the time series of the relative and absolute positions of the ten fingers w.r.t. some reference point;

different sport activities can be distinguished by the time series of some relevant physical quantities; all active sensors during a flight give time-changing values that form a time series. A repository of time series extracted from real-world data can be found in [3].

In its original formulation,model checking(MC) is the problem of verifying if a given formula is satisfied by a given model [12]. Usually, the model is the abstract representation of a system, in which the relevant properties become propositional letters, where the formula is written in a temporal logic and represents an interesting property. The prevailing adopted ontology for both the model and the logic is point-based: systems are represented in such a way that each state is a vertex on a Kripke model, atomic properties are descriptions of states, and the underlying logic is a point-based temporal logic, often LTL or CTL [32,33]. As interval-based temporal logics emerged as a possible alternative to point-based ones, the concept ofinterval temporal logic model checking(IMC) emerged with them. Halpern and Shoham’s interval temporal logic HS [18], which features one modality for each Allen relation [2], is the most representative interval-based temporal logic, and its model checking problem is the one that received the most attention. The problem of model checking HS formulæ has been formulated in two ways. On the one side, model checking for full HS, interpreted over finite Kripke structures according to a state-based semantics has been studied in [24,28]. The authors showed that, under the homogeneity assumption, which constrains a proposition letter to hold over an interval if and only if it holds over each component state, the problem is non-elementarily decidable (EXPSPACE- hardness has been later shown in [5]). Since then, the attention was brought to HS fragments, which are often computationally much better [5,6,25,26,27]. Also, the model checking problem for some HS fragments extended with epistemic operators has been investigated in [20,21]. The semantic assumptions for these epistemic HS fragments differ from those of [24,28] in two important respects, making it difficult to compare the two families of logics: formulæ are interpreted over the unwinding of the Kripke structure (computation-tree-based semantics) and interval labeling takes into account only the endpoints of intervals. The latter assumption has been later relaxed by making it possible to use regular expressions to define the labeling of proposition letters over intervals in terms of the component states [22]. On the other side, the problem of checkingfinite, linear, and fully representedinterval models (FIMC problem) against HS formulæ was formulated in [14], and its infinite, periodicalgeneralization was presented in [15] for a fragment of HS.

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These interval model checking strategies share the following elements: first, the object to be checked is abstracted in some way, and, second, the underlying temporal logic is acrisp(i.e., non-fuzzy) logic. We define multivariate time series model checking as the problem of checking if an interval temporal logic formula is satisfied by a finite multivariate time series. Time series are represented without abstraction, and we capture the intrinsic uncertainty carried by real-world data by checking formulæ of the fuzzy generalization of HS (FHS) [13], effectively introducing the fuzzy interval logic multivariate time series checking problem (TFIMC). On the one hand, fuzzy model checking has received very little attention in the literature, having been attempted in [16,31] for fuzzy generalizations of CTL; however, these frameworks are not directly comparable with the one we present here. On the other hand, our approach could be associated with the concept of probabilistic model checking (PMC) on Markov models, which has a large recent history in the literature (see [19], and references within); however: (i) fuzzy logics generalize probabilistic logics by having non-crisp both accessibility relations and atomic properties, (ii) probabilistic model checking is point-based, and (iii) Markov models are, as Kripke models, abstractions of the underlying systems. The TFIMC problem is the fuzzy, time series generalization of the FIMC problem. The major obstacle in solving the latter lies in the representation of the model, which may be exponentially succinct w.r.t. the size of the input (i.e., the pair model+formula). In [14] it has been proved that FIMC is polynomial, but the necessary pre-processing of the model has an im- pact on the overall complexity. Such an obstacle does not present itself in solving TFIMC, as time series cannot be succinctly represented, and, as a consequence, its complexity is lower than FIMC; however, the problem itself is conceptually not easier (in fact, it is slightly more difficult), and such a difference is hidden, so to say, in the representation of the model.

2 Preliminaries: Fuzzy Interval Temporal Logic

Crisp interval temporal logic.LetD=hD,≤i be a linearly ordered set. An interval overDis an ordered pair [x, y], wherex, y∈D andx < y. While in the original approach to interval temporal logic intervals with coincident endpoints were included in the semantics, in the recent literature they tend to be excluded except, for instance, in [4] where a two-sorted approach has been studied. If we exclude the identity relation, there are 12 different relations between two intervals in a linear order, often calledAllen’s relations [2]: the six relationsRA

(adjacent to, or meets), R_L (later than),R_B (begins),R_E (ends),R_D (during), and R_O (overlaps), depicted in Figure 1, together with their inverses R_X = (R_X)⁻¹, for each X ∈ {A, L, B, E, D, O}. We interpret interval structures as Kripke structures, with Allen’s relations playing the role of the accessibility relations. Thus, we associate a universal modality [X] and an existential modality hXiwith each Allen’s relationRX. For eachX ∈ {A, L, B, E, D, O}, theinverse of the modalities [X] andhXiare the modalities [X] andhXi, corresponding to the inverse relationR_XofRX. Halpern and Shoham’s logic, denoted HS [18], is a

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HS

hAi hLi hBi hEi hDi hOi

Allen’s relations

[x, y]RA[x⁰, y⁰]⇔y=x⁰ [x, y]RL[x⁰, y⁰]⇔y < x⁰ [x, y]RB[x⁰, y⁰]⇔x=x⁰, y⁰< y [x, y]RE[x⁰, y⁰]⇔y=y⁰, x < x⁰ [x, y]RD[x⁰, y⁰]⇔x < x⁰, y⁰< y [x, y]RO[x⁰, y⁰]⇔x < x⁰< y < y⁰

Graphical representation

x y

x0 y0

Fig. 1.Allen’s interval relations and HS modalities.

multi-modal logic with formulæ built from a finite, non-empty setAP of atomic propositions (also referred to as propositional letters), the classical propositional connectives, and a modal operator for each Allen’s relation, as follows:

ϕ::=⊥ |p| ¬ϕ|ϕ∨ϕ| hXiϕ.

In the above grammar, p∈ AP and X ∈ {A, L, B, E, D, O, A, L, B, E, D, O}.

The other propositional connectives and constants (e.g., →, and>), as well as the dual modalities, can be defined in the standard way (e.g., [A]ϕ≡ ¬hAi¬ϕ).

Given a formula of HS, its inverse formula is obtained by substituting every operator hXi with its inverse one hXi, and the other way around, for X ∈ {A, L, B, E, D, O}, while itssymmetricis obtained by substituting every op- eratorhXiwith its inverse onehXi, and the other way around, forX ∈ {A, L, O}, and everyhBi(resp.,hBi) with¯ hEi(resp.,hEi), and the other way around.¯

The semantics of HS is given in terms ofinterval modelsof the type:

M =hI(D), Vi,

where D is a linear order, I(D) is the set of all intervals over D, and V is a valuation function V : AP 7→ 2^I⁽^D⁾, which assigns to each atomic proposition p∈AP the set of intervalsV(p) on whichpholds. In this work, we are interested in finite structures and thus we restrict our attention to linear orders over finite domains. A finite domain of lengthnwill be denoted [n]. Thetruthof a formula ϕ on a given interval [x, y] in an interval model M is defined by structural induction on formulæ as follows:

M,[x, y]p if [x, y]∈V(p), forp∈AP; M,[x, y]¬ψ if M,[x, y]6ψ;

M,[x, y]ψ∨ξ if M,[x, y]ψor M,[x, y]ξ;

M,[x, y]hXiψ if M,[z, t]ψfor [z, t] s.t. [x, y]RX[z, t], forX ∈ {A, L, B, E, D, O, A, L, B, E, D, O}.

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During the past decades, several computational problems related to the logic HS have been studied, including the satisfiability problem, analyzed for the full logic in the original work by Halpern and Shoham [18], in which the authors prove that it is undecidable when the logic is interpreted in virtually all interesting classes of linearly ordered sets, and for various fragments (with different computational behaviours) in, among others, [1,8,9,10,23,29,30], the model checking problem, in [21,24,14], and, more recently, different knowledge extrac- tion problems, in [7,11].

Fuzzy interval temporal logic. Fuzzy HS was introduced in [13], where its satisfiability problem, along with certain expressive power problems, were studied. A formula of a fuzzy modal logic is usually evaluated in a Heyting Algebra.

AHeyting Algebra is a structure:

H= (H,∧,∨,→,0,1),

that is, a bounded distributive lattice with (non-empty) domainH, with internal operations∧(meet⁶) and∨(join), both commutative, associative, and connected by the absorption law, in which a partial order can be defined as:

αβ⇔α∧β =α⇔α∨β =β.

The symbols 0 and 1 denote, respectively, least and the greatest elements ofH.

In a Heyting algebra, therelative pseudo-complementofαw.r.t.β (akaHeyting implication), usually denoted asα→β, is defined as:

_{γ|α∧γβ},

and it exists for every αand β [17]. A Heyting algebra is said to be complete if for every subset S ⊆ H, both its least upper bound WS and its greatest lower boundVS exist. Typical realizations of Heyting algebras include the two- element Boolean algebra and the closed interval [0,1] in R. Given a complete Heyting algebra H with domain H, the fuzzy generalization of HS is defined starting with a domain Denriched with two functions:

<,e = :e D×D→H, and defining the structure:

De =hD,<,e =ie

as afuzzy linearly ordered setif it holds, for everyx, y,andz:

1. =(x, y) = 1e ⇔x=y (reflexivity of=);e 2. =(x, y) =e =(y, x) (symmetry ofe =);e 3. <(x, x) = 0 (irreflexivity ofe <);e

6 This is the classical nomenclature in lattice theory, and it should not be confused with Allen’s relationmeets, used in this paper.

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4. <(x, z)e <(x, y)e ∧<(y, z) (transitivity ofe <);e

5. <(x, y)e 0 &<(y, z)e 0⇒<(x, z)e 0 (transfer of<);e 6. <(x, y) = 0 &e <(y, x) = 0e ⇒=(y, x) = 1 (weak totality);e 7. =(x, y)e 0⇒<(x, y)e ≺1 (non-contradiction of <e over=).e

Observe that in the above formulæ we have used a meta-language with {&,⇒ ,⇔}in order to avoid confusion. Given a set of propositional letters AP and a complete Heyting algebra H, a well-formedfuzzy interval temporal logic (FHS, for short) formula is obtained by the following grammar:

ϕ::=α|p|ϕ∨ϕ|ϕ∧ϕ|ϕ→ϕ| hXiϕ|[X]ϕ,

whereα∈H,p∈AP, and, as in the crisp case,X ∈ {A, L, B, E, D, O, A, L, B, E, D, O}. We use¬ϕto denote the formulaϕ→0.

Given a fuzzy linearly ordered set, we can now define the set of fuzzy (strict) intervals inDe:

I(eD) ={[x, y]|<(x, y)e 0}.

Generalizing classical Boolean evaluation, propositional letters are directly evaluated in the underlying algebra, by defining a fuzzy valuation function Ve : AP ×I(De) → H that generalizes the crisp function V. Similarly, Allen’s relations now have a fuzzy behaviour, which is obtained by generalizing the original, crisp definition, and substituting every = with= and everye <with<:e

ReA([x, y],[z, t]) ==(y, z);e ReL([x, y],[z, t]) =<(y, z);e

ReB([x, y],[z, t]) ==(x, z)e ∧<(t, y);e ReE([x, y],[z, t]) =<(x, z)e ∧=(y, t);e ReD([x, y],[z, t]) =<(x, z)e ∧<(t, y);e

ReO([x, y],[z, t]) =<(x, z)e ∧<(z, y)e ∧<(y, t).e

Now, we say that anH-valued interval model(orfuzzy interval model) is a tuple of the type:

Mf=hI(eD),Vei,

where De is a fuzzy linearly ordered set that respects properties 1-7, andVe is a fuzzy valuation function. We interpret an FHS formula in a fuzzy interval model Mfand an interval [x, y] by extending the valuationVe of propositional letters as follows, where X∈ {A, L, B, E, D, O, A, L, B, E, D, O}and [z, t] varies inI(eD):

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Ve(α,[x, y]) =α;

Ve(ψ∧ξ,[x, y]) =Ve(ψ,[x, y])∧Ve(ξ,[x, y]);

Ve(ψ∨ξ,[x, y]) =Ve(ψ,[x, y])∨Ve(ξ,[x, y]);

Ve(ψ→ξ,[x, y]) =Ve(ψ,[x, y])→Ve(ξ,[x, y]);

Ve(hXiψ,[x, y]) =_

[z,t]

{Re_X([x, y],[z, t])∧Ve(ψ,[z, t])};

Ve([X]ψ,[x, y]) =^

[z,t]

{ReX([x, y],[z, t])→Ve(ψ,[z, t])}.

We say that a formula ϕ of FHS is α-satisfied at an interval [x, y] in a fuzzy interval model Mf=hI(eD),Vei, denoted M ,f [x, y]^α ϕ, if Ve(ϕ,[x, y])α. The formula ϕis α-satisfiable if and only if there exists a fuzzy interval model and an interval in that model where it is α-satisfied. A formula is satisfiable if it is α-satisfiable for someα∈ H,α 6= 0. A formula isα-valid if it is α-satisfied at every interval in every model, and valid if it is 1-valid. Observe that since a Heyting algebra, in general, does not encompass classical negation, and since our definition of satisfiability is graded, rather than absolute, the usual duality between satisfiability and validity does not hold anymore.

3 Fuzzy Time Series Checking

Multivariate time series. A temporal variable is a variable whose value changes over time. A time series is a set of temporal variables. They can be univariate, if only one temporal variable is involved, ormultivariate, otherwise.

In data science, a time series defined over a set of temporal variables (or temporal attributes)A={A1, . . . , A_m}is usually represented as ann×m matrix:

T =







a_1,1 a_1,2. . . a_1,m a_2,1 a_2,2. . . a_2,m . . . . a_n,1a_n,2. . . a_n,m





 .

We denote thedomainof an attributeA, that is, the set in whichAtakes values, by dom(A). We assume that each variable A_i that forms a multivariate series T has n values, and that there are no missing values, or that missing values are symbolized by placeholders; so, the lengthof a multivariate time seriesnis well-defined, as well as itswidthm. ByA(t), we denote the value of variableA at point t. An example of time series with m = 2 andn = 8 can be found in Figure 2; in this example, the variableA1(circles) represents the evolution of a patient’s temperature during the observed period, while the variableA2(stars) represents his/her blood pressure during the same period.

Problem definition. Let T be a time series of length n defined over a set of variablesA={A1, . . . , Am}. We define a set ofdecisionsS as:

S={A ./ a|A∈ A, a∈dom(A), ./∈ {≤, <,=, >,≥}},

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1 2 3 4 5 6 7 8 36

37 38 39 40 41 42

90 95 100 105 110 115 120

•

• •

•

• •

•

?

? ?

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Fig. 2.A multivariate time series with two variables.

and we set S as our set of propositional letters. Now, consider a time series T withndistinct points, and the finite domain [n]+ 1] obtained by adding a point 0 at the beginning of the series with undefined valuesa0,i, for eachi. After having fixed two concrete relations= ande <, we can form the set of intervalse I([n^+ 1]), and define a functionf to give truth values to decisions:

f :S ×I([n^+ 1])→H.

By means off, we allow a decision to beH-valuated on an interval. Interpreting a formula of FHS on a time series T simply consists of defining a H-valued interval model:

Te=hI([n^+ 1]),V , fi,e and of imposing that decisions are evaluated throughf:

Ve(Ai./ a,[x, y]) =f(Ai./ a,[x, y]).

Since the variablesAiare all undefined on 0, we may assume thatf is undefined on every interval of the type [0, y] as well. Given a time seriesT, an algebraH, a valueα∈H, and a formulaϕof FHS, thefuzzy interval logic multivariate time series checking problem (TFIMC) is the problem of establishing if it holds:

T ,e [0,1]αϕ.

In a way, we can say that the function f implicitly defines the domain of the algebra on whichTeis defined.

Concretizing a model: an example.Given a time seriesT, there are many ways to produce a concrete temporal model; some of them are more intuitive than others. In this example, we assume Hto be the closed interval [0,1] inR equipped with the relationminimum(as meets) andmaximum(as join), and we describe a concretization of f. The function f can be thought as a black-box function representing the domain-expert knowledge. In the example in Figure 2, for instance, one has to decide when temperature can be considered, say,over 38 degrees. One possible way is to definef, for a generic variableA, relation./,

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valuea, and interval [x, y], as the ratio between the number of pointsx≤t≤y that satisfyA ./ aandy−x+1. Similarly, we have to describe a concretization of

= ande <. Among the many ways that exist to do so, we fix a positive parametere h ∈ N (which can be thought of as an horizon), and define two parametric versions for= ande <e as follows: =eh(x, y) is always 0, except when|x−y|< h, in which case it is ^h−|x−y|_h ; instead,<e_h(x, y) is 1 wheny−x > h, it is 0 when y < x, and it is ^y−x_h when 0 ≤ y−x ≤ h. It is immediate to see that they satisfy axioms 1-7. Moreover, ifh= 1 our definition immediately reduces to the crisp definitions of = and<. Consider, again, the time series from Figure 2 and its corresponding modelTeobtained with the above fuzzy equality and ordering relations, and in which we assume, for the purpose of this example, h = 4. A possibly interesting property to be evaluated on this time series isstarting from day two, it is never true that blood pressure is low while temperature is high.

We can translate such a property as starting from day two, there is no interval in which blood pressure is low (i.e., less than or equal to 100) and temperature is high (i.e., greater than or equal to 38). The existence of such an interval is translated to FHS as:

hLi(A2≤100∧A₁≥38).

Among all witnesses of such a condition inT, we find the interval [5,e 6]:<(1,e 5) =

4

4, f(A1 ≥ 38,[5,6]) = ²₂, and f(A2 ≤ 100,[5,6]) = ²₂, so that Ve(hLi(A2 ≤ 100∧A1≥38),[0,1]) = 1.

Algorithm.We are ready to formalize the fuzzy time series checking algorithm.

Let Te =hI([n^+ 1]),V , fe i be a model based on some complete algebraH, ϕa formula of FHS, and α ∈ H. Algorithm 1 is the adaptation of Emerson and Clarke’s classical CTL algorithm to the interval, fuzzy case, and it returnstrueif and only if the value ofϕon the (auxiliary) interval [0,1] is greater than or equal to αin Te. In Algorithm 1, we use the symbol◦ ∈ {∨,∧,→}to denote a logical symbol, and the symbol• to denote its algebraic corresponding one. Unlike the crisp case, every sub-formula must be checked on every interval, because, in the fuzzy case, any two intervals [x, y] and [z, t] may be related by any relationR

Xe. The auxiliary data structure L can be thought of as a hash table indexed by three elements, namelyψ, x, y, that is, a sub-formula, and two points. Accessing L may be considered to have constant time complexity. Formulæ, classically represented as binary trees, can be pre-processed in order to identify repeating sub-formulæ, so that the main cycle of Algorithm 1 can be implemented in an efficient way. It is worth observing that such a solution implicitly assumes that nis a reasonably low value; for very high values ofn, a different solution should be designed for the dimension ofL to be manageable.

Complexity. Let Te = hI([n^+ 1]),V , fie be a model based on m temporal attributes, each withndistinct points, and letkbe the length of the input formula ϕ. Since the length of the time series grows as time passes, while its width remains unchanged as well as the property to check, we can assume that m, k =o(n),

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Algorithm 1Fuzzy interval logic multivariate time series checking algorithm.

1: functionCheck(T , ϕ, α)e 2: L=∅

3: forψ∈ sub(ϕ) in increasing length orderdo 4: if ψ=β, withβ∈H then

5: for[x, y]∈I([n^+ 1])do

6: L(ψ,[x, y]) =β

7: if ψ=A ./ athen

8: for[x, y]∈I([n^+ 1])do 9: L(ψ,[x, y]) =f(A ./ a,[x, y]) 10: if ψ=τ◦ξthen

11: for[x, y]∈I([n^+ 1])do

12: L(ψ,[x, y]) =L(τ,[x, y])•L(ξ,[x, y]) 13: if ψ=hXiτ then

14: for[x, y]∈I([n^+ 1])do

15: s= 0

16: for[z, t]∈I([n^+ 1])do

17: s←s∨(R

Xe([x, y],[z, t])∧L(τ,[z, t]))

18: L(ψ,[x, y]) =s

19: if ψ= [X]τ then

20: for[x, y]∈I([n^+ 1])do

21: s= 1

22: for[z, t]∈I([n^+ 1])do

23: s←s∧(R

Xe([x, y],[z, t])→L(τ,[z, t]))

24: L(ψ,[x, y]) =s

25: returnL(ϕ,[0,1])α

that is, that there are much less temporal variables and much less sub-formulæ than there are distinct points. Thus, we can express the size of the input as O(n). Also, we can assume that the join (∨), meet (∧), and Heyting implication (→) operations take constant time inn, and that each call tof takes timeO(n) (in the worst case scenario, in fact, each call to f requires exploring an interval with n points). The most external cycle is executedO(n) times. In the worst- case scenario, during each executionψis a modal formula. Since there areO(n²) intervals inTe, the complexity of the modal case isO(n⁴). Therefore, the entire algorithm runs inO(n⁵).

Theorem 1. The TFIMC problem can be solved in polynomial time by a deter- ministic algorithm.

4 Conclusions

In this paper we first defined, and then solved, the multivariate time series fuzzy interval logic checking problem. Despite its simplicity, the interest in this prob-

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lems lies in the fact that multivariate time series are ubiquitous in certain areas of data science and learning, but they have never before been linked to the classical model checking problem. Yet, we believe that many interesting properties can be formulated, and therefore checked, on a time series, and that the recently introduced fuzzy interval temporal logic FHS is a suitable language to do so. As future work, our intention is to design and test an efficient implementation of our algorithm, and to explore further interactions between model checking and learning, specifically, symbolic learning of FHS formulæ over time series.

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